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$$$z - 10 \sin{\left(x \right)}$$$ の $$$x$$$ に関する積分
入力内容
$$$\int \left(z - 10 \sin{\left(x \right)}\right)\, dx$$$ を求めよ。
解答
項別に積分せよ:
$${\color{red}{\int{\left(z - 10 \sin{\left(x \right)}\right)d x}}} = {\color{red}{\left(\int{z d x} - \int{10 \sin{\left(x \right)} d x}\right)}}$$
$$$c=z$$$ に対して定数則 $$$\int c\, dx = c x$$$ を適用する:
$$- \int{10 \sin{\left(x \right)} d x} + {\color{red}{\int{z d x}}} = - \int{10 \sin{\left(x \right)} d x} + {\color{red}{x z}}$$
定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=10$$$ と $$$f{\left(x \right)} = \sin{\left(x \right)}$$$ に対して適用する:
$$x z - {\color{red}{\int{10 \sin{\left(x \right)} d x}}} = x z - {\color{red}{\left(10 \int{\sin{\left(x \right)} d x}\right)}}$$
正弦関数の不定積分は$$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$です:
$$x z - 10 {\color{red}{\int{\sin{\left(x \right)} d x}}} = x z - 10 {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$
したがって、
$$\int{\left(z - 10 \sin{\left(x \right)}\right)d x} = x z + 10 \cos{\left(x \right)}$$
積分定数を加える:
$$\int{\left(z - 10 \sin{\left(x \right)}\right)d x} = x z + 10 \cos{\left(x \right)}+C$$
解答
$$$\int \left(z - 10 \sin{\left(x \right)}\right)\, dx = \left(x z + 10 \cos{\left(x \right)}\right) + C$$$A